Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

8. Sharing the gains from marriage 345
the accumulated marginal changes, as we move up the stable assignment
profile to marriages with higher incomes. Because of the interaction in
traits, the change in the marital contribution depends on the income of
the spouse that one gets. Note that marginal increases in x0 or y0 have
no effect on u(x) or v (y) , respectively, because the marginal persons with
these incomes are just indifferent between marrying and remaining single.
In marriages that involve individuals from the bottom of the male and
female income distributions, members of the larger sex group typically have
higher income. Thus, if r>1 andallmenaremarried,themeninthelowest
quality matches have almost no income, while their wives have strictly
positive income. The wife receives her utility as single h (0,y0) and the
husband receives the remaining marital output. If r =1, the allocation
in the lowest quality match is indeterminate and, consequently, there is a
whole set of possible sharing rules that differ by a constant of integration.
8.2.2 A tractable specification
Let us now slightly generalize our previous approach by assuming that male
incomes x are distributed on a support [a, A] according to some distribution
F and female incomes y are distributed on a support [b, B] according to
some distribution G; the assumption of different supports for men and
women is useful for empirical applications. We introduce now a simplifying
assumption, namely that the output function h depends only on total family
income. That is,
h(y, x) =H(y + x), (8.14)
with H (0) = 0. This assumption makes sense in our transferable utility
setting, since under TU a couple behaves as a single decision maker. Note
that basically all examples of intrahousehold allocation with TU given in
Chapter 3 satisfy this property.
Under this assumption, hy(y, x) =hx(y, x) =H 0 (y + x), and assortative
matching requires hxy (y,x) =H 00 (y + x) > 0, sothatH is increasing and
convex. As above, we let ψ (x) (resp. φ (y)) denote the income of Mr. x’s
(Mrs. y’s) spouse. Finally, we maintain the convention that a single person
with income s (= x, y) achieves a utility level H (s).
We are interested in how changes in the sex ratio and the distributions
of income of the two sexes affect the allocation rule that is associated with
a stable matching. In this analysis, we shall distinguish between two issues:
(i) the shape of the allocation rule in a cross section of marriages - that is,
how do the shares vary as we move up the assignment profile to couples
with higher incomes, and (ii) changes of the allocation rule as parameters
of the marriage market, such as the sex ratio or the male and female income
distribution, change.